The idea of using hyperbolic localization to obtain a localization formula in cohomological DT theory was first formulated by Balazs Szendroi section 8.4 of [1]}. It was applied in section 6 of [2]} and in the section 8.3 [3]}, where it was used in a specific example of framed representations of quiver with potential. In [4]}, Timo Richarz proved the commutativity of the hyperbolic localization with the vanishing cycles functor in greater generality. We have used this result to establish the formula (REF ) for any critical locus of a potential in [5]}, and the extension of this result to d-critical structures was suggested to us by Richard Thomas.
